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Testing an in-line digital holography ’inverse method’
for the Lagrangian tracking of evaporating droplets in
homogeneous nearly isotropic turbulence
Delphine Chareyron, Jean-Louis Marié, Corinne Fournier, Jérôme Gire,
Nathalie Grosjean, Loïc Denis, Michel Lance, Loïc Méès
To cite this version:
T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s
New Journal of Ph ysics
Testing an in-line digital holography ‘inverse method’
for the Lagrangian tracking of evaporating droplets
in homogeneous nearly isotropic turbulence
D Chareyron1,4, J L Mari ´e1,5, C Fournier2, J Gire2, N Grosjean1, L Denis3, M Lance1 and L M ´e `es1
1Laboratoire de M´ecanique des Fluides et d’Acoustique UMR5509,
Ecole Centrale de Lyon—CNRS—Universit´e Claude Bernard Lyon 1— INSA Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France
2Laboratoire Hubert Curien UMR5516, CNRS—Universit´e Jean-Monnet,
18 rue Pr Benoit Lauras, F-42000, St Etienne, France
3Centre de Recherche Astronomique de Lyon UMR5574,
CNRS—ENSL—Universit´e Claude Bernard Lyon 1, France E-mail:jean-louis.marie@ec-lyon.fr
New Journal of Physics 14 (2012) 043039 (26pp)
Received 21 July 2011 Published 26 April 2012 Online athttp://www.njp.org/
doi:10.1088/1367-2630/14/4/043039
Abstract. An in-line digital holography technique is tested, the objective being to measure Lagrangian three-dimensional (3D) trajectories and the size evolution of droplets evaporating in high-Reλstrong turbulence. The experiment
is performed in homogeneous, nearly isotropic turbulence (50 × 50 × 50 mm3)
created by the meeting of six synthetic jets. The holograms of droplets are recorded with a single high-speed camera at frame rates of 1–3 kHz. While hologram time series are generally processed using a classical approach based on the Fresnel transform, we follow an ‘inverse problem’ approach leading to improved size and 3D position accuracy and both in-field and out-of-field detection. The reconstruction method is validated with 60 µm diameter water droplets released from a piezoelectric injector ‘on-demand’ and which do not appreciably evaporate in the sample volume. Lagrangian statistics on 1000 reconstructed tracks are presented. Although improved, uncertainty on the depth positions remains higher, as expected in in-line digital holography. An additional filter is used to reduce the effect of this uncertainty when calculating the droplet
4Present address: Laboratoire de Physique de l’ENSL, UMR5672 CNRS, 46 All´ee d’Italie, 69364 Lyon Cedex 07,
France.
5
Author to whom any correspondence should be addressed.
velocities and accelerations along this direction. The diameters measured along the trajectories remain constant within ±1.6%, thus indicating that accuracy on size is high enough for evaporation studies. The method is then tested with R114 freon droplets at an early stage of evaporation. The striking feature is the presence on each hologram of a thermal wake image, aligned with the relative velocity fluctuations ‘seen’ by the droplets (visualization of the Lagrangian fluid motion about the droplet). Its orientation compares rather well with that calculated by using a dynamical equation for describing the droplet motion. A decrease of size due to evaporation is measured for the droplet that remains longest in the turbulence domain.
Contents
1. Introduction 2
2. Experimental setup 4
2.1. Turbulence facility . . . 4
2.2. Injection system . . . 5
2.3. Freon conditioning circuit. . . 8
2.4. Digital in-line holography setup—tracking method . . . 9
3. Image processing of holograms 11 3.1. ‘Inverse problem’ approach algorithm . . . 11
3.2. Wake masking for robust estimation . . . 12
4. Results 13 4.1. The case without appreciable evaporation: water droplets . . . 13
4.2. Evaporating freon droplets . . . 18
4.3. Inferring Lagrangian fluid motion about a droplet . . . 21
5. Conclusion 23
Acknowledgments 24
References 24
1. Introduction
affected by inertia and so differs from that of fluid particles. But as the droplets evaporate, they become smaller and smaller compared to the relevant scales of the turbulence and, at the limit, can approach the behaviour of passive tracers. It is difficult to investigate this coupling between evaporation and turbulence in real situations, where it is combined with other effects (coalescence, breakup, wall interactions, etc). Besides, the Eulerian measuring techniques that are used in most studies on dispersion–evaporation are not best suited for providing a complete understanding of physical mechanisms involved [3]. Our objective is to obtain experimental information on this coupling from a Lagrangian point of view in a simpler, fundamental situation, where the important parameters of the problem (the Lagrangian time scale of turbulence, the response and evaporation times of the droplets) can be controlled. Practically, the experiment consists in releasing a dilute train of monodispersed droplets in a homogeneous, nearly isotropic strong turbulence and measuring their size evolution in conjunction with their trajectories and spatial dispersion. This paper focuses on the measuring technique that has been selected for this study.
Most Lagrangian techniques are recent: about 15 years old or less, and have essentially been used to investigate the dispersion of particles in turbulent flows. A detailed survey can be found in the review paper by Toschi and Bodenschatz [4]. They are based on various principles such as 3D particle tracking velocimetry (3D PTV) with four cameras [5, 6], optical imaging with silicon strip detectors [7–10], extended laser Doppler anemometry (extended LDA) [11], ultrasonic Doppler tracking [12] and digital holography [13–15]. Compared to the other techniques mentioned, the main advantage of digital holography is that it provides information on size, which is obviously crucial for evaporation studies and caused us to select it. The method that has been chosen is an in-line digital holography setup. This setup is well suited to Lagrangian measurements in flows with low concentrations of particles [16, 17], which is typically the case here (see section 2.2). Moreover, it is insensitive to vibrations and easy to implement, two important points for our operating conditions. The in-line digital holography method has already been proved to be successful in studying the turbulent dispersion of slightly buoyant droplets in homogeneous, nearly isotropic turbulence with high-speed cameras [13, 18]. However, the rms velocity fluctuations in this case are low, of the order of 10 cm s−1,
and no phase change takes place. The situation that most resembles ours is that explored by Lu et al [14]. These authors tested an in-line single high-speed camera method on droplets dispersing in strong turbulence within a box [19] and compared its accuracy to that of a stereo setup with two high-speed cameras. They showed that the in-line method is capable of measuring particle size and trajectories in this type of flow, and that it has certain advantages compared to the stereo method: simplified setup and calibration, need of less laser power, no particle-matching between cameras and a deeper sample volume (even for small numerical aperture experiments). From their results, obtained with a standard hologram reconstruction method, a number of limitations of in-line digital holography can be listed.
• The spatial resolution along the optical axis is poor (about 100 times lower than in the transverse directions). This yields a poor estimation of particle acceleration along this axis. • The sample volume is small in the transverse directions: 17.6 mm × 13.2 mm (for a setup
without magnification, the lateral field of view is restricted to the width of the sensor). • The sizing accuracy along the tracks is found to be of the order of ±10% with a simple
As emphasized by many authors, there are several ways to push back these limits. One of them is to use improved algorithms for digital hologram processing. An ‘inverse problem’ approach dedicated to 3D reconstructions of objects with simple parametric shapes (most notably, droplets modelled by spheres) has recently been proposed [20–22] and tested both on simulated data and on a jet of monodispersed droplets. This approach proves particularly promising to address some of the main issues encountered in in-line holography today. The proposed algorithm improves depth positioning accuracy, provides particle diameters with sub-micrometer accuracy, eliminates bias close to the hologram border and increases the volume of the reconstructed domain [23]. The purpose of this work, therefore, is to apply this approach to our highly turbulent flow and show that it retains its performance in real conditions. The method is tested with water droplets (the case exhibiting no appreciable evaporation) and freon droplets (the case with potential evaporation). The paper is organized as follows. First we describe the experimental setup, including the system generating the turbulence, the droplet injectors, the conditioning freon circuit and the digital in-line holography setup. Then the algorithms used to process the holograms are detailed. Finally, the results obtained in the two cases are presented and interesting aspects of the method for the Lagrangian evaporation study are discussed. The conclusion summarizes the most important points of this work.
2. Experimental setup 2.1. Turbulence facility
The experiment has been performed in homogeneous isotropic turbulence whose detailed characterization can be found in [24]. The facility, inspired by the original idea of Hwang and Eaton [19], consists of an ‘open’ turbulence chamber displayed in figure 1. Turbulence is produced by the mixing of six large synthetic jets, in an ambient atmosphere. The jets are created by six loudspeakers which are mounted on the lateral sides of a 0.9 m2cubic frame, so as
to form the vertices of a regular octahedron. Unlike the chamber designed by Hwang and Eaton there are no walls, which has some advantages for the present study, in particular as confinement is almost non-existent, no significant secondary motions take place outside the quasi-isotropic turbulence. As a result, the probability of droplets that leave this domain of interest re-entering the domain is almost nil. This is an essential requirement for experiments where the coupling between the evaporation process and turbulence dispersion is studied. From a practical point of view, the absence of walls is favourable for optical techniques such as digital holography, as an important source of noise, such as fringes created by interference on the glass windows, is avoided. Each loudspeaker is driven with sine waves with the same 42 Hz frequency and phase. The balance between the six loudspeakers is optimized at the beginning of the experiment by adjusting the driving signals, so as to obtain zero average velocities and equal fluctuations in the X and Y directions. This is checked by LDA measurements at the centre of the domain (X = Y = Z = 0).
The measurements presented in the next sections have been taken in turbulence with a Reynolds number based on the Taylor scale Reλ= 293. The properties of this turbulence
were measured by particle image velocimetry (PIV), as described in [24]. The domain where turbulence is homogeneous, nearly isotropic, is a cube whose side LTD is about 50 mm. The
Figure 1. (a) Schematic diagram of the turbulence-generating system. The homogeneous isotropic domain is approximately a cube with LTD= 50 mm sides
located at the centre; (O, X, Y, Z ) denote the axes of reference for the domain; (b) layout of the holographic setup in the plane X –Z ; (O, x, y, z) denote the axes of reference attached to the optical setup, with z being the optical axis. (c) Locations of the droplet generator: 1, the case of water; 2, the case of freon.
Table 1.Turbulence characteristics.
Twice the Dissipation Taylor Reynolds Turnover Integral Kolmogorov kinetic energy rate microscale number time length scale length scale time
q2(m2s−2) ǫ (m2s−3) λ (mm) Reλ TE(ms) 3 (mm) ηk(mm) τk(ms)
2.12 5.78 5.2 293 56 47 0.16 1.60
error estimates
4–5% 20% 11% 13% 20% 15–20% 5% 9%
the turbulence are summarized in table 1. The Lagrangian time scale TL, characteristic of the
exponential decrease of the Lagrangian gas velocity autocorrelation, could not be measured directly. It was estimated both from the eddy-turnover time TE≈ 0.5 q2/(3ǫ) and from the
Kolmogorov time τk= (ν/ǫ)0.5. Referring to [25–27], TL/TE≈ 0.76–0.78 and TL/τk≈ 20–24
for the Reλinvestigated here. This yields a value of TL in the 32–45 ms range.
2.2. Injection system
For the evaporation of droplets to be investigated from a Lagrangian point of view, one must be able to measure their diameter along the 3D trajectories. To show that the in-line holographic method allows this measurement with a good degree of confidence, it was tested with monodispersed droplets, well controlled in size.
(a) (b)
Figure 2. (a) Photograph of the MJ-AT-01 injector MicroFab Technologies. 1: Glass capillary; 2: metallic case; 3: connections for the piezoceramic; 4: threaded end. (b) PDF water droplet’s diameter as measured by PDA without turbulence (thin line) and with turbulence (thick line).
orifice diameter (1), embedded in a piezoceramic tube. This kind of injector has the advantage of being miniaturized, but also of enabling two types of functioning. It can work either in a commonly known ‘continuous droplet jet-mode’ or in a ‘droplet-on-demand mode’ [28, 29]. The on-demand mode has been adopted here because it provides monodispersed droplets with a stable size over long periods of time (a few hours) and which are pretty well reproduced from one run to the next. The droplets thus generated have approximately the same diameter as that of the glass capillary, i.e. about 60 µm.
This diameter has been chosen so that the response time τd of the droplets is around TL
for the fluids used in the study, which is the condition for significant coupling to take place. The fluids are demineralized water, which does not evaporate appreciably on the residence time scales, and freon R114 whose boiling point is 3.8◦C under atmospheric pressure. The
water was used as a test fluid to show that trajectories and diameters could be measured with high accuracy with our digital holography technique. In particular, it was verified that, since there is no appreciable evaporation in this case, the diameters along each track remain constant. The typical response times are of the order of 10–20 ms, which yields a Stokes number Stk= τd/τk≈ 6.5–12.5. Droplets are generated at a frequency of 500 Hz. The monodispersed
Table 2.Droplet characteristics.
Average diameter rms diameter
Flow d(µm) σd(µm)
Without turbulence 61.7 0.5
With turbulence 60.1 2.6
The droplets are injected above the homogeneous isotropic domain into a point on the vertical Y -axis (X = Z = 0 mm) located at a distance of 40 mm from the centre of the domain (location 1 in figure 1(c)). This location has been adopted because it respects the symmetry of the loudspeakers and minimizes the flow disturbance. The distance to the centre has been selected by compromise. On the one hand, the velocity of the droplets entering the turbulence domain must not be too high compared with the turbulent velocity fluctuations. On the other hand, their momentum must be sufficient to allow them to penetrate the domain.
Y = 40 mm is reasonable to fulfil both conditions. Simple considerations based on the
droplet dynamical equation show that a 60 µm water droplet injected there penetrates the domain at Y = 25 mm with a velocity of the order of magnitude of the turbulence fluctuations and reaches its settling velocity inside the domain. If the distance is too great most of the droplets are expelled outside the domain and in this case samples are definitely too few to make acceptable statistics during the length of time when loudspeakers are stable, typically 60 min [24]. This kind of difficulty arises from the non-confinement and the fact that the number of droplets delivered by the injector is limited. It is not encountered in closed turbulence chambers [14,30], where particles or drops are introduced with their settling velocity in much higher concentrations and remain captured inside the box. With the compromise adopted here, 4% of the droplets released penetrate the turbulence domain.
The penetration rate has been measured by using the interferometric laser imaging for droplet sizing (ILIDS) technique. It has been shown that it was possible to increase the penetration rate up to 13% by injecting from the lateral side (X = −90 mm, Y = 20 mm,
Z = 0 mm) facing one of the loudspeakers, so that the turbulent jet arriving there drives them
more easily into the turbulence domain (location 2 in figure1(c)). However, this location being comparatively more intrusive, it was not used for the holographic tests with water droplets. PDA measurements of the droplet size have been made at the centre of the domain with the injector at location 1. We see in figure 2(b) that the size distribution is slightly broader than when measured immediately at the outlet of the injector without turbulence (table 2). As was checked by ILIDS, this broadening is not a measuring bias associated with the location of the PDA volume relative to the injector: at the outlet or far from it. It is more likely caused by the fact that turbulence strongly destabilizes the jet of droplets close to the tip of the capillary and, by doing so, influences the droplet formation.
As the droplets are injected at a frequency of 500 Hz, their average number Nd in the
turbulence domain is estimated to be 500τr, where τr is their mean residence time inside this
domain. τr is about 13.3 ms (see section 4.1), which yields Nd≈ 7. This number is consistent
with the number of droplets recorded on average per hologram (figure7(a)). The droplet number density nd= Nd/L3TDis thus very low, of the order of 0.06 cm−3, and far below the limit at which
1 (a) (b) 2 3 4 5 6 7 8 9 10
Figure 3.Temperature conditioning for the injection of freon R114. (a) Cooling circuit; 1: TC50E Huber immersion cooler; 2: adjustment of the cooling set-point temperature; 3: serpentine exchanger; 4: refrigerant bath; 5: recirculating cooling tube, 6: peristaltic pump; 7: thermocouple for measuring the temperature at injection. (b) Injection circuit; 8: R114 pressurized tank; 9: DL-BP1 high-precision pressure-reducing valve for cryogenic fluids; 10: Mastercool pressure reducing valve.
shadow density criterion of Royer [32], the limit for obtaining high-quality holograms with our parameters (the diameter of the droplets and size of the sample volume) is nd= 50 cm−3.
2.3. Freon conditioning circuit
Freon R114 is stored in a high-pressure tank and first had to be liquefied before being injected at atmospheric pressure. This requires decreasing the temperature below 0◦C, and for that reason,
a special supply device was designed to achieve the phase change and transport the R114 in its liquid form up to the nozzle. Basically, the device consists of two separate circuits (figure 3). The cooling circuit (a) necessary for the phase change and for keeping the R114 liquid includes an immersion cooler (1), a serpentine exchanger (3) and a refrigerant bath (4), whose cooling set point is adjusted by a controller (2). The refrigerant of the bath (R290) is recirculated by a peristaltic pump (6) in a 8 mm diameter silicon tube (5) up to the glass capillary, from where it then returns to the bath. The temperature of the R114 is measured at the entrance of the capillary by a thermocouple (7). The injection circuit (b) includes the R114 pressurized tank (8), a high-precision reducing valve for cryogenic fluids (9) and a mastercool pressure-reducing valve (10). The R114 is depressurized down to atmospheric pressure, enters the cooling bath where it is liquefied and is driven up to the nozzle. The injection pressure is adjusted by (9) with an accuracy of 5 mbar. The set-point temperature of the cooling bath is typically −10◦C, while the temperature of the R114 just before injection is −5◦C. The size of the R114
Figure 4.PDF of the freon droplets’ initial diameters. Statistics on 63 droplets. Mean diameter: 243 µm; standard deviation: 33 µm.
k k k ,y ,z ,d Pk k(x ) 1380 mm S l ≅ l0≅580 mm CMOS 12 mm focal length LASER S 0 z y x Diverging lens of
Figure 5.Schematic diagram of the in-line holography setup.
2.4. Digital in-line holography setup—tracking method
The turbulence domain is illuminated by a Millenia IIs Spectra Physics solid laser (see figure5), emitting coherent light at λlaser= 532 nm. The optical axis z of the source is positioned in the
plane X –Z of the domain, at an angle of 40◦ to the X -axis (see figure1(b)). This angle proves
to be necessary for good optical access to the domain of interest. The beam at the laser outlet is expanded by a 12 mm diverging lens located 1380 mm from the plane where water droplets are injected ((x, y, z = 0) in figure1(b)). The uniformity of the obtained wave front was found satisfactory and thus no spatial filter was added. The sensor of the camera is placed in front of the source, 580 mm from the injection plane.
The slight beam divergence (about 4◦) that results causes a magnification, improving
somewhat compensated for by the capability of the ‘inverse problem’ approach to detect out-of-field droplets [21]. The magnification factor depends on the longitudinal coordinate zk of the
particle k. This geometrical magnification Gk [33] is given by
Gk =
lS+ l0
lS− zk
, (1)
where lS, l0denote the distances from the point source to the calibration plane (O, x, y) and from
the calibration plane to the sensor, respectively (figure5). During the reconstruction procedure, this varying magnification must be taken into account to estimate the actual coordinates. To estimate this factor more accurately, we use a graduated calibration target (Reticle Edmunds Optics, F56-347) positioned in the injection plane. In this plane, the measured magnification is
G0= 1.42 ± 0.005, and Gk can be expressed as a function of G0, l0 and zk in the form
Gk =
G0
1 − (G0− 1)zlk
0
. (2)
The relationship between the actual longitudinal coordinate zk and that measured by digital
holography zmesk is zk = −l0+ zmesk + l0mes Gk , l0= l0mes G0 . (3)
Thus Gk can be expressed from G0, l0mes and zmesk as
Gk = G0+
zmesk lmes
0
(G0− 1). (4)
The reconstructions presented in section4account for this non-uniform magnification.
to handle incoming and outgoing droplets of the imaging field. Also, the positions zkt along
the beam axis were restricted to the range (−25 mm < zkt <+25 mm), which corresponds to
droplets located inside the turbulence domain. A very few pairing errors are generated by this tracking method. They occur exceptionally when two droplets with crossing trajectories move close to each other. In these cases, a three-frame minimum acceleration criterion [34] was used to find the right pairing.
3. Image processing of holograms 3.1. ‘Inverse problem’ approach algorithm
3D reconstruction of digital holograms is classically performed in two steps. The first step is based on a numerical simulation of the optical reconstruction. A 3D image volume is obtained by computing the diffracted field in planes located at increasing distances z from the hologram. Different techniques to simulate diffraction have been proposed (Fresnel transform [17], fractional Fourier transform [37,38] and wavelets transform [39,40]). The second step consists of segmenting, i.e. localizing and sizing each object in the obtained 3D image. The best focusing plane for each object has to be detected. Various criteria have been suggested in the literature. The limitations encountered in such approaches have been well identified [23]. The most restrictive are the poor depth accuracy and the limited field of view due to border effects.
In this framework, Soulez et al [20, 21] proposed an algorithm for parametric objects (objects whose shape can be described by a few parameters) 3D reconstruction, avoiding 3D volume reconstruction. This hologram analysis amounts to diffraction-pattern detection. The expression of the diffraction pattern of a spherical particle with a given size and location is well known in the far-field case [41] and easy to compute (direct problem). In our experiment, particles are spherical; therefore the diffraction pattern depends on only four parameters: their 3D location (xk,yk,zk) and diameter (dk). The inverse problem consists in reconstructing the
set of droplets (locations and sizes: {xk,yk,zk,dk}k=1,...,n) whose computed diffraction pattern
best matches the real hologram. Finding this set is equivalent to a global optimization problem in which the difference between the diffraction model and the data (hologram) is minimized through a penalty function. Practically, this global optimization problem is solved by an iterative algorithm that alternates the coarse location of the droplets, refinement by local optimization and subtraction of the pattern from the hologram (cleaning step) [20].
As the diffraction pattern of each droplet is affected by the patterns due to the others, repetition of the processing over the residual images (obtained after cleaning) enables droplets with faint diffraction signatures to be detected and located, in particular those out of the field of view. This was shown both on synthesized and on real holograms (jet of 90 µm diameter droplets). The benefits of the ‘inverse problem’ approach are discussed in detail in [23]. This approach gives improved particle location precision, in particular along the depth dimension, and increases the size of the effective transverse field of view. A notable drawback of the ‘inverse problem’ approach is the longer processing time required compared to conventional hologram reconstruction methods.
as for back-propagation; local optimization for sub-pixel accuracy is costly. On a standard desktop computer (3 GHz processor, single thread), the detection and location of a single particle in a hologram (with field expansion) requires about 5 min, and a series of 1000 holograms about 2 weeks of computations. For holograms with an average of five droplets like here, the reconstruction without out-of-field detection with an ‘inverse problem’ approach is about 15 times longer than with standard hologram processing techniques.
Accelerating the ‘inverse problem’ approach is currently ongoing work. Several ways are possible for that. Holograms can be processed in parallel using batch processing on several cores/nodes. In the case of holograms with many more particles, detecting several particles at a time is an efficient way to save computations. Finally, except for the discovery of new particles, the processing time can be significantly decreased if the tracking algorithm is used for the coarse detection of the particles on the next frame.
3.2. Wake masking for robust estimation
In the evaporating case, a wake image can be seen on the diffraction patterns of each droplet. These wakes, whose nature and physical interest are discussed in detail in section 4.2, disturb the signal. In this section, we show how to reduce the wake’s influence on the measurements.
A distinctive feature of the digital holography technique is that the signal of an object is non-localized (i.e. is spread over the whole sensor). This important feature leads to the 3D reconstruction of out-of-field objects [21] or of a part of a hologram (considering some missing data). The ‘inverse problem’ approach algorithms take into account a confidence degree on each pixel by using a weighted mask depending on the pixel noise variance. In the case when the noise is assumed to be constant over all known pixels, the weighted mask can be viewed as a binary mask on the data: equal to 1 if the pixel is measured and to 0 otherwise. This mask weight allows wake disturbance on the freon hologram to be reduced.
Freon droplets are spherical objects whose hologram formation model is well known: the model consists of a diffraction pattern with radial symmetry. It is a chirp function whose amplitude is modulated by a Jinc function. The object’s depth is encoded in the pattern frequency modulation, while the radius information is encoded in the amplitude modulation. The wake has a nearly constant thickness (about 60 pixels) and an origin located at the pattern centre. Therefore, the wake disturbs the low frequencies of the signal (recorded on the small-radius rings) more than the high frequencies (recorded on large-small-radius rings). Note that it has been shown [42] that the high frequencies of the signal are of lower magnitude than the low frequencies, but participate in accurate measurement (as they are more sensitive to changes in
z and are recorded on more pixels). In order to reduce the influence of wakes on the hologram analysis, we chose to refine our measurements by repeating the ‘inverse problem’ approach algorithm using a mask suppressing the low-frequency part of the signal whose signal-to-noise ratio is weak due to the wake. The mask therefore consists of discs centred on the pattern centre (see in figure6)
A first iteration of the algorithm gives us a rough estimate of the droplets parameters (xk,yk,zk,dk). A refinement step of this measurement is carried out using a mask of weight
0 on a disc centred on the approximate xk,yk positions of the droplets and radius equal to the
(a) (b)
Figure 6. (a) An example of a binary mask. (b) The mask applied to the hologram.
(a) (b)
Figure 7. (a) A typical hologram of 60 µm water droplets, acquired with the high-speed 1280 × 1024 NAC K4 digital camera. (b) An example of 3D reconstructed tracks of 60 µm water droplets dispersing inside the turbulence domain.
4. Results
4.1. The case without appreciable evaporation: water droplets
A typical hologram of water droplets can be seen in figure7(a).
About 1000 3D tracks of droplets inside the turbulence domain have been reconstructed (see the illustration in figure 7(b)). This required the processing of about 10 000 holograms. The length of these tracks ranges from 2–3 ms (2–3 time steps) to 40 ms, with an average of about 13.3 ms. This average, of the order of 8 Kolmogorov time τkand 0.3–0.4 integral time TL,
represents the mean residence time τr of the droplets inside the turbulence domain. This means
ranges from 2–3 to 60 mm, with an average of about 13.7 mm. This average is of the order of 86 Kolmogorov length ηkand 0.3 integral length 3. Only a few tracks, among the longest, have
a length comparable to 3 and sometimes a bit larger.
The uncertainties on the lateral positions, δx, δy, and the depth position, δz, estimated from
the holograms reconstruction have been determined theoretically by calculating the standard deviation σx, σy, σz of the estimates [42]. Following this approach, the accuracy attainable on
the droplet positions is at best equal to the Cramer–Rao lower bounds [43, 44]. These bounds depend on the parametric model of the signal used in the reconstruction algorithm (diffraction patterns) and on the level of noise in the hologram. These lower bounds have been calculated for values of the parameters close to our experimental conditions. The number and the size of the pixels 1ξ , the fill factor, the imaging distance, the size d and the position of the droplets are taken into account in the calculation. We thus find that accuracy is at best σx ≃ σy≃ 3 µm, i.e.
≃ d/20 ≃ 1ξ/14, for the lateral positions, and σz≃ 60 µm, i.e. ≃ d ≃ 31ξ , for the depth
po-sition. In reality, accuracy on positions is probably not as good due to the presence of noise not taken into account in the theoretical estimates (background noise and noise due to interference between the diffraction patterns of the particles). The fact that uncertainty is higher in the depth direction is a classical problem with a single camera in-line holography setup with a small numerical aperture [45]. However, this uncertainty can be decreased to a value approaching that of the lateral directions by filtering the trajectories, as shown below. To compute the velocity and acceleration statistics, the positions along the tracks must be differentiated once or twice, respectively, and this would severely amplify the noise. Velocities and accelerations have been computed both with and without filtering. The filtering method is the same as that proposed by Mordant et al [9]. In practice, the spatial coordinates of the particle are convolved with a Gaussian kernel, which is differentiated once to obtain the velocity and twice for the acceleration. This convolution is equivalent to low-pass filtering of the trajectories to filter out the noise resulting from the position uncertainties. In the x and y directions where accuracy on positions is high, the effect of such filtering is very limited. This is illustrated in figures 8(a) and (c), where the non-filtered x components of the acceleration and velocity (dashed lines) fit the ones obtained by filtering (solid lines) well. In contrast, the beneficial effect of the filtering algorithm on the accelerations and velocities is clearly visible along the longitudinal
z-direction, where the uncertainties on positions are much higher (figures8(b) and (d)).
The PDFs of the three fluctuating velocity components computed with filtering are plotted on a semi-log scale in figure 9(a). The statistics were obtained from 7000 samples and each distribution was normalized by its standard deviation. The distributions for the three components are approximately Gaussian with standard deviations which are close to each other: σvx = 0.90 m s−1, σvy = 0.80 m s−1, σvz= 0.78 m s−1. This Gaussian behaviour, which is
expected for fluid particles in homogeneous isotropic turbulence [46,47], was also reported by Snyder and Lumley [48] in the case of inertial particles (glass and copper particles) dispersing in a wind tunnel. It is noted that the data exhibit some scattering in the [−3, 3] standard deviation range, even along the x and y directions, where the accuracy on location is high. This scattering is attributed to the low number of samples (1000 tracks) and the fact that PDFs are likely not converged. In comparison, the number of tracks currently used for Lagrangian statistics in other works with high Reλ [12, 49, 50] is typically of the order of 5000–30 000,
Figure 8. Velocity and acceleration components of a water droplet before and after filtering. v denotes the velocity and a the acceleration. x components are plotted on the left-hand side (panels (a) and (c)), while z components are plotted on the right-hand side (panels (b) and (d)). Open circle: raw data; solid line: filtered data.
Figure 10.Velocity auto-correlation functions of water droplets.
The PDFs of the three acceleration components are presented in figure 9(b). As expected, without filtering, the variance of the z component was found to be higher than in the other two directions due to the lower accuracy on the longitudinal position. The initial rms value was σaz= 88 m s−2 for the longitudinal component, compared to σax = 46 m s−2, σay= 47 m s−2,
for the transversal ones. The filtering step reduces the variance of the longitudinal position estimation, leading to comparable values of acceleration in each direction: σax= 43 m s−2,
σay= 44 m s−2, σaz = 55 m s−2. As is seen in the figure, the PDFs of the three components
normalized by their rms values superpose rather well. Compared with a Gaussian with the same variance (solid line), they seem to have an exponential form. But the tails do not extend very far, as is reported for this Reλ[8]. This can be attributed to various combined reasons. The number
of tracks being small compared to that required in acceleration Lagrangian statistics, the scarce strong events, whose probability is smaller than 10−3, are hardly measured. These strong events
are even more difficult to see because the residence time of the droplet is short; hence long trajectories are not numerous. Also, the Stokes numbers of the droplet are larger than unity, so they do not respond rapidly to turbulence fluctuations, thus causing the turbulence effect to be filtered [51]. Finally, the time between samples (1 ms), typically of the order of Kolmogorov’s time (1.6 ms), is still too long to measure these strong accelerations.
The Lagrangian velocity autocorrelation coefficients were computed by taking into account the bias corrections introduced by Mordant et al [12]. This bias arises from the fact that the tracks are not of the same length and that generally the longest ones correspond to the slowest particles. To limit this bias, the authors proposed to normalize the coefficients calculated at time τ by the velocity variance of recorded tracks whose duration is longer than τ . For the x component of the Lagrangian velocity this reads
Rx x(τ ) =
1 σ2
vx(τ )
h(Vx(t+ τ )Vx(t )i, (5)
where hi denotes the average over the tracks that contribute to Rx x(τ ), and σv2x(τ )is the variance
Figure 11. (a) Typical example of a water droplet trajectory and (b) of the Lagrangian time evolution of its diameter. Symbols from black to white: increasing time.
in the turbulence domain. However, the behaviour at long times must be considered with great care because the number of tracks is too low for the statistics to be meaningful. Fitting the beginning of the plot by a decaying exponential leads to a characteristic time of 40 ms, in good agreement with the estimations of the Lagrangian time given in section 2.1. Note that it is extremely difficult to evaluate the error on the unbiased correlation function thus estimated because it combines the velocity statistics and the lengths of the tracks. Referring to [52], it is reasonable to state that the variance of this estimation should vary as 1/√N (τ ), where N (τ ) is the number of tracks used for the calculation at τ . On this basis, the error on Rx x(τ )is expected
Figure 12.Distribution of the relative diameter deviations along the Lagrangian tracks. The relative diameter deviation is the difference between the diameter along a track and the mean diameter, divided by the mean diameter. hdi = 58.9 µm; rms standard deviation ±1.6%.
the real signal-to-noise ratio was not taken into account in this theoretical estimation, accuracy on diameters is in reality not as good.
To evaluate the real accuracy on the diameter measurements along the tracks, the same method as in Lu et al [14] was used. This consists in computing the distribution of the relative diameter deviations along the Lagrangian tracks. For each track, this deviation is the difference between the diameter d measured at each time step and the mean diameter hdi along that individual track, divided by this mean diameter, that is, (d − hdi)/hdi. The result for the full dataset is displayed in figure12.
The remarkable point is that deviations do not exceed 2% for the 1000 tracks reconstructed in the domain, showing that accuracy on the diameter is close to the value estimated theoretically. As a comparison, in a turbulence box and with initial polydispersed droplet spray (60–120 µm), Lu et al [14] obtained a distribution with a relative deviation of the order of ±10% with maxima of the order of 20%. The setup and the accuracy of the method being validated in the non-evaporating case, freon droplets were then injected.
4.2. Evaporating freon droplets
Figure 13.Holograms of freon droplets recorded at 1 kHz. The time between two successive images is 2 ms. The droplet marked with an arrow moves toward the left, while its wake is at each instant aligned with the relative velocity fluctuations seen by the droplet.
Figure 14. Distribution of relative freon droplet diameter deviations along Lagrangian tracks (statistics on 63 tracks). The relative diameter deviation is the difference between the diameter along a track and the initial diameter d0, divided
by the initial diameter.
inertial behaviour, and so, do not follow the fluids trajectories. The Lagrangian fluid motion about a droplet is therefore visualized. This result is consistent with the size of the droplets injected. As mentioned in section 2.3, we did not succeed in obtaining stable injection modes because of some instabilities in the process of freon liquefaction and hoarfrost problems at the capillary tip. As a result, the droplets released in this case (figure4) are polydispersed with an average diameter of about 243 µm, much higher than that of the water droplets. Their response time is typically of the order of τd= 230 ms, while the Stokes number is Stk= τd/τk≈ 144.
Hence, they are strongly inertial. Perhaps a more sophisticated injection system, such as that used for instance by Gouge and Fisher (1997) [53] to jet cold xenon droplets in another context, would help to solve the problem.
The second remarkable result is, whatever the nature of the wakes (temperature and/or concentration), their lifetime is sufficient to follow their 2D spatial evolution (images 7–15 in the top right-hand corner). This provides a way to investigate how the topology of the concentration/temperature field is built, which is a particularly important issue regarding evaporation and combustion applications.
The most important information that sheds some light on the nature of the wake is the distribution of the relative diameter deviations along the Lagrangian tracks (figure 14). Due to the difficulty encountered in injecting the fluid, the number of tracks reconstructed is only 63. The distribution remains centred around zero, with deviations which do not exceed 2%, as in the case of water. It is deduced that on average practically no evaporation takes place on the time scales that the freon droplets reside in the sample volume and that the wakes are essentially thermal wakes, made up of the air cooled by the R114 freon droplets at a temperature close to 0◦C. This result is not surprising. As the droplets are strongly inertial, they mostly follow ballistic trajectories, resulting in a residence time that is too short (11.7 ms) for evaporation to become significant. This is confirmed by a simple estimation of the heating time τh necessary for the vaporization state to be reached. The initial conditions at the droplet
0 0.01 0.02 0.03 0.04 210 215 220 225 230 t(s) d( µ m) (a) (b)
Figure 15. (a) The trajectory of the freon droplet that remains longest in the turbulence domain. Time increases from black to white. (b) Evolution of its diameter as a function of time.
out accurately. Supposing that the temperature is the same as that inside the injector: T0≈ 0◦C,
the droplet is isolated in an atmosphere at T∞≈ 25◦C at rest and the temperature within the
droplet is homogeneous at each instant (infinite conductivity model), then the heating time can be estimated by [54] τh≈ − ρlCld02 12λ ln Tsat− T∞ T0− T∞ , (6)
where ρl= 1528 kg m−3is the density and Cl= 0.96 kJ kg−1K−1 the specific heat of the liquid,
λ = 0.96 mW m−1K−1the thermal conductivity of the air, d
0 the initial diameter of the droplet
and Tsat= 3.8◦C the saturation temperature in these working conditions. For R114 droplets with
an initial diameter of 243 µm, this yields τh≈ 45 ms, which is much greater than the average
residence time in the domain.
However, some evaporation is measured on one of the longest trajectories inside the field of view (figure15(a)). As can be noted in figure15(b), the diameter along the trajectory decreases by about 10 µm in 35 ms, which corresponds to an evaporation rate km= 1.29 × 10−7m2s−1.
This evaporation rate is much smaller than the theoretical value kth= 6.9 × 10−7m2s−1that can
be estimated by using the classical steady-state approach of Spalding’s law [55]. This suggests that the temperature of the droplet has not reached the saturation temperature and that the evaporation process is not maximum. The concentration of vapour at the surface is likely to be low and even in that case, the image wake should be mostly caused by a gradient of temperature.
4.3. Inferring Lagrangian fluid motion about a droplet
Supposing, for the sake of simplicity, that the main forces acting on it reduce to gravity and the drag force, this equation is written as (see, for instance, [56])
dud dt = 1 − 1 γ g +3CD(Red) 4dγ |u − ud|(u − ud), (7) where γ is the ratio of densities ρl/ρa, ud the droplet velocity, u the air velocity and Red
the Reynolds number based on the droplet diameter and the relative velocity ur= u − ud. The
Reynolds number for this size of droplet is in the range 0 6 Red6800, so the drag coefficient
is well described by the empirical correlation
CD(Red) =
24
Red
(1 + β Reαd), (8)
where β = 0.15 and α = 0.687. The projection of this equation along the three directions of the holographic setup yields the system
τdu˙dx = urx(1 + β Redα), (9)
τdu˙dy(1 − g) = ury(1 + β Reαd), (10)
τdu˙dz= urz(1 + β Reαd), (11)
where τd= ρld2/18µa and the symbol ( ˙ ) denotes the time derivative. By summing the square
of equations (9)–(11), one obtains
τd2(u˙dx2+ ˙udy2+ ˙udz2) = u2r(1 + β Re α d)
2
. (12)
τd, ˙udx2, ˙udy2 and ˙udz2have been calculated at each time step from the reconstructed trajectory
(figure 15(a)) and the measured diameter (figure 15(b)), while the unknown of the system
u2
r has been determined by finding the value solution of equation (12). Once u2r has been
determined, Red is known and the three components of the relative velocity can be readily
s 4 0 0 . 0 = t s 2 0 0 . 0 = t s 0 = t t = 0.011 s s 9 0 0 . 0 = t s 6 0 0 . 0 = t s 7 1 0 . 0 = t s 4 1 0 . 0 = t t = 0.021 s t = 0.024 s t = 0.028 s t = 0.030 s
Figure 16.Time sequence images showing the evolution of the wake of a freon droplet in the x–y-plane and that of the relative velocity in this plane seen by the droplet (white arrow), calculated with equations (9) and (10) and the Lagrangian data. The solid line on each image represents the trajectory of the droplet and the black arrow its velocity. From t = 0.009 to 0.021 s, the droplet is out-of-field: its location and diameter are still measured thanks to the off-frame particle detection advantages of the inverse method.
5. Conclusion
and by measuring their size evolution along their 3D trajectories from holograms taken with a single high-speed camera. The holograms have been reconstructed with an algorithm based on an original ‘inverse problem’ approach [20] that offers several advantages. It allows the detection of droplets located out of the field of view, a point that has been confirmed here. Lateral dimensions of the sample volume (40 × 40 × 140 mm3) are about 2.5 times greater than when one is restricted to conventional hologram reconstruction and thus are large enough to investigate the turbulence domain. This approach is also known to improve the accuracy of the droplet depth position and diameter. Its main drawback is a longer processing time compared to conventional methods.
The gain in position and diameter accuracy have been checked with 60 µm water droplets, which do not appreciably evaporate on the residence time scales in the sample volume. Uncertainty on the depth position expected for our operating conditions (60 µm for this drop diameter) is 20 times that on the transversal positions (3 µm or about 1 pixel size), whereas it is typically of the order of 110 times (≈ d/λlaser) with a standard reconstruction algorithm [45].
But in reality, depth–position accuracy is not so good, due to the presence of noise. This appears on the PDFs of Lagrangian accelerations obtained from 1000 reconstructed trajectories, where we note that the rms value of the longitudinal component is twice the value of the transverse ones. Depth uncertainty can be decreased to a value approaching that of the transverse directions by using a filtering algorithm, of the type proposed by Mordant et al [9]. With filtering, the PDFs of the three acceleration components superpose pretty well, as expected in an isotropic flow. Accuracy on diameters has been estimated to be ±1.6% around their average value of 58.9 µm by computing their variance along each trajectory. Tests on evaporation have been performed with R114 freon droplets. Their residence time in the turbulence domain proved, on average, to be too short for evaporation statistics. This is explained by some injection problems resulting in larger droplets (240 µm), with ballistic behaviour. However, a decrease in diameter has been measured on the droplet having the longest residence time. Very interestingly, a wake image created by a temperature gradient is present on each droplet hologram. This thermal wake originates from the air cooled by the droplets and is approximately aligned with the instantaneous relative velocity seen by the droplet. Its orientation, which visualizes the Lagrangian fluid motion about the droplet, compares rather well with that calculated by using a dynamical equation for describing the droplet motion. The space evolution of these wakes provides a way to investigate how the topology of the temperature/concentration field is built, which is a particularly important issue with regard to evaporation and combustion applications.
Acknowledgments
This work was supported by the r´egion Rhˆones-Alpes and ANR-BLAN-07-1-192604. The authors thank Romain Volk from ENSL and Mickael Bourgoin from LEGI for useful discussions. The particle positioning and sizing algorithm used in this paper was implemented in Yorick (http://yorick.sourceforge.net), a free data processing language written by D Munro.
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